A Net Present Value Approach in Developing Optimal Replenishment Policies with Allowable Shortages for a Product Life Cycle

In this paper, we utilize a net present value approach to devise replenishment policy with allowable shortages in the finite planning horizon by considering a general, time-varying, continuous, deterministic demand function for a product life cycle.We consider four possible alternatives for the inventory problem in the shortage models. (1) It starts with an instant replenishment and ends with zero inventories, (2) It starts with an instant replenishment and ends with shortages, (3) It starts with shortages and ends with zero inventories, or (4) It starts and ends with shortages. We investigate the optimal number of inventory replenishments, the corresponding optimal inventory replenishment time points and the shortage time points and analytically compare these four inventory models based on minimizing the total relevant cost. A complete search procedure is developed to find the optimal solution by employing the properties derived in this paper and the well-known Nelder–Mead algorithm. Also, several numerical examples are carried out to illustrate the features of these models by utilizing the search procedure developed in this paper. Then various models are analytically compared, and identified the best alternative among them based on minimizing total relevant costs.